I am trying to work through an exam review before Friday. I have figured out every problem except for one. It's driving me crazy. Was hoping someone could help me to figure it out...?
What does it mean for a function to be continuous on a set?
My choices:
1. A function is continuous on a set S if for every ϵ>0\nobreak, there is δ>0\nobreak such that |f(x)-f)c)|<ϵ\nobreak whenever x, c ∈\nobreakS and |x-c|<δ\nobreak
2. A function f:S→R is continuous on the set S if for every sequence {xn} in S, the sequence {f(xn)} converges
3. A function f:S→R is continuous at the point c∈S if for every ϵ>0, there is δ>0 such that |f(x)-f(c)|<ϵ whenever x ∈ S ad |x-c|<δ if f continuous at all c ∈ S then f is continuous on set S
4. A function f:S→R is continuous on the set S if the left limit of f is equal to the right limit of f at every point of S
5. A function f:S→R is continuous on the set S if f is continuous at every cluster point of S
I am leaning towards #3
Any help is appreciated
What does it mean for a function to be continuous on a set?
My choices:
1. A function is continuous on a set S if for every ϵ>0\nobreak, there is δ>0\nobreak such that |f(x)-f)c)|<ϵ\nobreak whenever x, c ∈\nobreakS and |x-c|<δ\nobreak
2. A function f:S→R is continuous on the set S if for every sequence {xn} in S, the sequence {f(xn)} converges
3. A function f:S→R is continuous at the point c∈S if for every ϵ>0, there is δ>0 such that |f(x)-f(c)|<ϵ whenever x ∈ S ad |x-c|<δ if f continuous at all c ∈ S then f is continuous on set S
4. A function f:S→R is continuous on the set S if the left limit of f is equal to the right limit of f at every point of S
5. A function f:S→R is continuous on the set S if f is continuous at every cluster point of S
I am leaning towards #3
Any help is appreciated